Section 4: Powers of an Element; Cyclic Groups For elements of a semigroup (S, ), the definition of positive integer exponents is clear: For x in S and n in Z +, x n = x x x, where there are n factors, i.e., n x s starred together this n-fold star is meaningful because associativity says that, no matter how parentheses are put in to divide it into n 1 2-fold stars, the result will always be the same. In particular, x 1 = x. If the semigroup has a (two-sided) identity e, then you can probably guess what an exponent of 0 means: x 0 = e. If S is really a group, i.e., every element also has an inverse, then we can make sense of negative exponents; in fact, there are technically two reasonable definitions for x n when n Z + : Is it the inverse of x n, or the n-th power of x 1 is x n equal to (x n ) 1 or is it (x 1 ) n?. Fortunately, these two results turn out to be equal [the text leaves this as an exercise for the student, and so will I]. Warnings: Fractional exponents are usually meaningless in the context of groups. Also, one of the familiar rules of exponents does not hold, unless the group is abelian: If we start with (x y) 2 = x 2 y 2 (for elements x, y of some group), i.e., x y x y = x x y y, then cancellation shows that y x = x y. So for a general, not-necessarily-abelian group we can only hope to prove the other two rules of exponents: Prop: If (G, ) is a group, them for all x in G and m, n in Z, we have x m x n = x m+n and (x m ) n = x mn = (x n ) m. If G is abelian, then for any x, y in G and n in Z, we have (x y) n = x n y n. Pf: For x m x n = x m+n, suppose first that m, n are both positive. Then x m x n = (x x x) (x x x) where the first set of parentheses has m x s and the second has n of them, so the result is starring together a total of m + n x s, i.e., x m+n. Suppose one of them is 0, say m = 0; then the equation to be proved is x 0 x n = x0 + n, but because x 0 = e and 0 + n = n (no matter whether n is positive, 0 or negative), both sides are x n and the result follows. The text does the case where both m, n are negative and where m < 0 and n > 0, so let s do the case where m > 0 and n < 0; so that k = n is a positive integer. Then x m x n = x m x k = x m (x 1 ) k, the value of which depends or the relative values of k, m, telescoping the stars x x 1 in the middle of the product: if k < m, then x m (x 1 ) k = x m k = x m+n ; if k = m, then x m (x 1 ) k = e = x 0 = x m k = x m+n ; and if k > m, then x m (x 1 ) k = (x 1 ) k m = x (k m) = x m+n. Thus, in all cases, x m x n = x m+n. The proof that (x m ) n = x mn is also an exercise for the student. (It is not necessary to break up in cases for m, only for n.) Because multiplication of integers is commutative, the equality x mn = (x n ) m follows from the other one. Finally, suppose G is abelian, x, y G and n Z. Suppose first that n is positive. Then (x y) n = x y x y x y is a star of n x s and n y s, and with the commutative property we can rearrange to put all the x s first and all the y s last, so that it becomes x n y n. If n = 0, then (x y) n = e and x n y n = e e = e, so they are equal. Finally, if n is negative, say n = k, then x k y k = (x y) k so e = x k (x y) k y k = x k y k (x y) k, so (x y) k = x k y k, i.e., also in this case, (x y) n = x n y n.// Notation: From now on, a general operation will be denoted, not by, but by juxtaposition, writing xy instead of x y. Occasionally, if the operation is commutative, we will denote it by +; in this case the inverse of x is denoted by x, and powers become multiples: x + x + + x (with n terms) is denoted nx. So the rules of exponents become (m + n)x = mx + nx, m(nx) = 1
(mn)x = n(mx), and because we are assuming commutativity, n(x + y) = nx + ny. In our basic examples of operations, addition and multiplication in R, most of the multiples nx and powers x n (except for the identities, 0 and 1 respectively) get larger and larger as the n gets larger and larger. The counterexample is powers of 1, which alternate between 1 and 1. But in other groups, the powers (multiples) of elements can cycle through any number of other elements before repeating. For example, in (Z 5, ), the multiples of 1 are Similarly, the powers of i in (C {0}, ) are 1 1 + 1 = 2 1 + 1 + 1 = 3 1 + 1 + 1 + 1 = 4 1 + 1 + 1 + 1 + 1 = 0 1 + 1 + 1 + 1 + 1 + 1 = 1 1 + 1 + 1 + 1 + 1 + 1 + 1 = 2 etc. i, i 2 = 1, i 3 = i, i 4 = 1, i 5 = i, i 6 = 1, etc. Terminology: Let S be a subset of a group G. We say that S is closed under the operation on G if, for all x, y in S, xy is also in S, i.e., the restriction of the operation on G to S is an operation on S. And S is closed under inverses if, for all x in S, x 1 is also in S. If a nonempty subset S of G is closed under the operation and inverses, then S is a group in its own right, called a subgroup of G. We ll study more general subgroups later, but for this section we are interested in a specific kind of subgroup: Def and Prop: Let x be an element of a group G. The set x of all powers of x is closed under the operation on G and under inverses, so it is a subgroup of G, called the cyclic subgroup generated by x. (i) If there is no positive integer n for which x n = e, then we say x has infinite order; in symbols, o(x) =. In this case the function ϕ : Z G : n x n is a one-to-one function with range x, and it shows that, as a group, x behaves just like Z. (ii) If there is a positive integer n for which x n = e, then the smallest such n is called the order of x, denoted o(x). In this case, for o(x) = n, the function ψ : Z n G : k x k is a one-to-one function with range x, and it shows that, as a group, x behaves just like Z n. Pf of whatever isn t a definition in this statement: The subset x is closed under the operation because x m x n = x m+n ; it is closed under inverses because (x n ) 1 = x n. So it is a subgroup of G. In the case where there is no positive power of x that is equal to e, it is clear that ϕ has range x. to see it is one-to-one, suppose x m = x n where m n; then x m n = e, so we must have m n = 0, i.e., m = n. In the case where o(x) = n <, we want to show first that ψ has range all of x, i.e., every power of x is equal to x r for some r between 0 and n 1 (inclusive): Given the power x m, longdivide m by n: m = qn + r where q, r Z and 0 r < n; then x m = x qn+r = (x n ) q x r = e q x r = x r. 2
And to see that ψ is one-to-one, suppose x r = x s where 0 r s < n; then x s r = e, but 0 s r < n and the choice of n means s r = 0, i.e., r = s. Behaves just like means that adding in Z or Z n corresponds to multiplying in G: In the case of infinite order, we have ϕ(n + m) = x n+m = x n x m = ϕ(n)ϕ(m), so the operations in the groups x and Z are essentially identical. And in the case of order n, let s, t be elements of Z n and long-divide the integer s + t by n: s+t = nq +r. Then s t = r, and ψ(s)ψ(t) = x s x t = x s+t = x n q +r = (x n ) q x r = x r = ψ(s t). So the operations in the groups x and Z n are essentially identical.// Here is a diagram of what the operations are essentially identical means in the infinite-order case: Z Z + Z ϕ ϕ ϕ G G op G If we start with any pair of integers in the upper left, going across and then going down gives the same result as going down (side-by-side) and then going across. Ex: In the group GL(3, R), let Then A 2 = so A = {A, A 2, I} behaves just like Z 3 : A =., A 3 = = I ϕ : Z 3 A : 0 I, 1 A, 2 A 2 0 1 2 2 1 1 2 0 2 2 0 1 I A A 2 I I A A 2 A A A 2 I A 2 A 2 I A If x n = e, then (x 1 ) n = (x n ) 1 = e 1 = e, and vice versa, so o(x), the smallest n for which x n = e, is the same as o(x 1 ), and if one is infinite, so is the other. We can say more when we are talking about finite orders, but the proofs suddenly involve a lot of Math 250 results. Before we give the proof, let s take an example, looking at the orders of the elements of Z 12 : 0 has order 1 the identity in any group is the only element of order 1. 1 has order 12: we need to add 12 copies of 1 to get 0. 2 has order 6: 2(2) = 4, 3(2) = 6, 4(2) = 8, 5(2) = 10, 6(2) = 0. The idea that works for 2 also works for the other divisors of 12: 3 has order 4, 4 has order 3, 6 has order 2. How many copies of 5 will we need to add to get 0? I.e., in Z, how many 5 s will we need to add to get a multiple of 12? Well, because 5 has no factor in common with 12, we only get n(5) = m(12) if n is divisible by 12; so 5 has order 12. The same is true of the other numbers less than 12 and relatively prime to it: 7 and 11. 3
1(8) = 8, 2(8) = 4, 3(8) = 0; i.e., in Z, a multiple of 8, n(8), is a multiple of 12 only if n makes up the factor 3 of 12 that is not in 8. In the equation n(8) = m(12), divide both sides by the gcd of 12 and 8: n(2) = m(3); because 3 is relatively prime to 2, it must divide n. So o(8) = 3 = 12/4 = 12/ gcd(12, 8). And the same idea works for 9 and 10: In Z, 1(9) = 9, 2(9) = 18, 3(9) = 27, 4(9) = 36, a multiple of 12: o(9) = 4 = 12/3 = 12/ gcd(12, 9); and 6(10) = 60 is the smallest common multiple of 10 and 12: o(10) = 6 = 12/2 = 12/ gcd(12, 10). So the idea seems to be that, in Z n, the order of an element k is n/ gcd(n, k). If this is right, it should translate to any cyclic group x where o(x) = n. We need two things from Math 250, i.e., from arithmetic in Z: the idea of long division, and the fact that, if a divides a product bc and gcd(a, b) = 1, then a c. The text proves these in great detail, but because they are now done in Math 250, I will assume we know them. Prop: Suppose the group element x has finite order n. Then: (i) For any integer m, x m = e if and only if n m; and (ii) For any integer k, o(x k ) = n/ gcd(n, k). Pf: (i) Of course if n m, say m = nd, then x m = (x n ) d = e d = e. Conversely, suppose x m = e, and long-divide m by n: m = qn + r where q, r Z with 0 r < n. Then we have e = x m = x qn+r = (x n ) q x r = e q x r = x r. But n was the smallest positive power of x that is e, so r, which is less than n, cannot be positive, i.e., it must be 0. Thus, n divides m = qn. (ii) Because gcd(n, k) is a factor of k, k/ gcd(n, k) is an integer, so (x k ) n/ gcd(n,k) = (x n ) k/ gcd(n,k) = e k/ gcd(n,k) = e. So suppose there is an integer m for which (x k ) m = e; in view of (i), it is enough to show that n/ gcd(n, k) divides m: We know that x km = e, so n km, say km = ns where s Z. Dividing both sides by gcd(n, k) gives (k/ gcd(n, k))m = (n/ gcd(n, k))s. But k/ gcd(n, k) and n/ gcd(n, k) have no factors in common we have divided out all the common factors so they are relatively prime; so the fact that n/ gcd(n, k) divides the product (k/ gcd(n, k))m but is relatively prime to the first factor means that it must divide the second factor. Thus, n/ gcd(n, k) divides m, as required.// Def: If a group G includes an element x for which all the elements of G are powers of x, i.e., x = G, then G is called a cyclic group, and x is called a generator of G. If o(x) =, we still call x a cyclic group, even though nothing is cycling. For any group G, the cardinality G is called the order. If G = x is cyclic, then G = o(x). Because the powers of an element all commute with each other, a cyclic group is abelian. But there are abelian groups that are not cyclic: The text gives the examples of (Q, +) (assume x is a generator; then x/2 is in Q, but it is not an integral multiple power of x, a contradiction) and the Klein Four-Group V = {e, a, b, c} with the operation [inspired by Z 2 2 under addition modulo 2] e a b c e e a b c a a e c b b b c e a c c b a e [ (0, 0) (1, 0) (0, 1) (1, 1) (0, 0) (0, 0) (1, 0) (0, 1) (1, 1) (1, 0) (1, 0) (0, 0) (1, 1) (0, 1) (0, 1) (0, 1) (1, 1) (0, 0) (1, 0) (1, 1) (1, 1) (0, 1) (1, 0) (0, 0) 4 ].
Every element is its own inverse, so no element has order 4. 5